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- W2949418251 abstract "We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the edge probability, and write $p=frac{1+vep}{2(n-1)}$ for some $vepin R$. In Borgs et al., Random subgraphs of finite graphs: I. The scaling window under the triangle condition, Rand. Struct. Alg. (2005), and in Borgs et al., Random subgraphs of finite graphs: II. The lace expansion and the triangle condition, Ann. Probab. (2005), the size of the largest connected component was estimated precisely for a large class of graphs including H(2,n) for $vepleq Lambda V^{-1/3}$, where $Lambda > 0$ is a constant and $V=n^2$ denotes the number of vertices in H(2,n). Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when $vepgg (log{V})^{1/3} V^{-1/3}$, then the largest connected component has size close to $2vep V$ with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of $p$ are supercritical. Barring the factor $(log{chs{V}})^{1/3}$, this identifies the size of the largest connected component all the way down to the critical $p$ window." @default.
- W2949418251 created "2019-06-27" @default.
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- W2949418251 date "2008-01-10" @default.
- W2949418251 modified "2023-09-27" @default.
- W2949418251 title "Random subgraphs of the 2D Hamming graph: the supercritical phase" @default.
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